An overlapping domain decomposition method for parametric Stokes and Stokes-Darcy problems via proper generalized decomposition

TL;DR

This paper introduces a non-intrusive, efficient domain decomposition method combined with proper generalized decomposition for creating local surrogate models of parametric Stokes and Stokes-Darcy flows, enhancing computational speed and flexibility.

Contribution

The paper develops a novel DD-PGD framework that constructs physics-based local surrogate models for coupled flow problems, compatible with various discretization schemes and avoiding auxiliary basis functions.

Findings

Demonstrates high accuracy and robustness of DD-PGD in numerical tests.

Shows significant reduction in computation time compared to high-fidelity solvers.

Provides a seamless, non-intrusive approach adaptable to different discretizations.

Abstract

A strategy to construct physics-based local surrogate models for parametric Stokes flows and coupled Stokes-Darcy systems is presented. The methodology relies on the proper generalized decomposition (PGD) method to reduce the dimensionality of the parametric flow fields and on an overlapping domain decomposition (DD) paradigm to reduce the number of globally coupled degrees of freedom in space. The DD-PGD approach provides a non-intrusive framework in which end-users only need access to the matrices arising from the (finite element) discretization of the full-order problems in the subdomains. The traces of the finite element functions used for the discretization within the subdomains are employed to impose arbitrary Dirichlet boundary conditions at the interface, without introducing auxiliary basis functions. The methodology is seamless to the choice of the discretization schemes in space, being compatible with both LBB-compliant finite element pairs and stabilized formulations, and the DD-PGD paradigm is transparent to the employed overlapping DD approach. The local surrogate models are glued together in the online phase by solving a parametric interface system to impose continuity of the subdomain solutions at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the entire overlap and without solving any additional physical problem in the reduced space. Numerical results are presented for parametric single-physics (Stokes-Stokes) and multi-physics (Stokes-Darcy) systems, showcasing the accuracy, robustness, and computational efficiency of DD-PGD, and its capability to outperform DD methods based on high-fidelity finite element solvers in terms of computing times.